I am self-learning numerical analysis from the text Introduction to Numerical analysis, by Suli and Mayers. On page 11, the authors prove a local version of the contraction mapping theorem. The first couple of steps of the proof are not clear to me. Would someone be able to help explain, how he arrives at these steps.
Theorem. Suppose that $g$ is a real-valued function, defined and continuous on a bounded closed interval $[a,b]$ for all $x \in [a,b]$. Let $\xi = g(\xi) \in [a,b]$ be a fixed point of $g$ (whose existence is ensured by Theorem 1.2) and assume that $g$ has a continuous derivative in some neighbourhood of $\xi$ with $|g'(\xi)|<1$. Then, the recursive sequence $(x_k)$ defined by $x_{k+1}=g(x_k)$, $k \geq 0$, converges to $\xi$ as $k \to \infty$ provided that $x_0$ is sufficiently close to $\xi$.
Proof.
Let $\xi \in (a,b)$. By hypothesis, there exists $h>0$, such that $g'$ is continuous in the interval $[\xi - h, \xi + h]$.
Since, $|g'(\xi)| < 1$, we can find a smaller interval $I_\delta = [\xi - \delta,\xi+\delta]$, where $0<\delta\le h$, such that $|g'(x)|\le L$ in this interval, with $L < 1$.
I really don't understand what theorem or corollary has been used to arrive at the above assertion.